Mathc: Calculus, Gnuplot.

Algebra linear in language c :

matrxf.zip : You can work with fractions.

addm, subm, multm, powm, smultm, transpose, trace,det, minor, mminor,
cofactor, mcofactor,adjoint,inverse(adjoint), inverse(gaussjordan),
inverse(identity matrix), gauss, gaussjordan, LU.

norm, distance, innerproduct,coldim, rowdim, rank, nullity, leastsqrs

matrxg.zip : You can also work with integers.

matrxh.zip : The print, copy, rand, ... functions.

matrxj.zip : System of equations with the free variables.

In this section, the size of the matrices, are randomly selected
by the computer, but you can selecte the size if you want.

Verify with numeric applications:

mtrxaa.zip : How to use the basic functions.

when the size of the matrices are randomly selected by the computer.

mtrxab.zip : the properties of matrix arithmetic.

A+B = B+A
(A+B)+C = A+(B+C)
(AB)C = A(BC)
A(B+C) = AB+AC
(B+C)A = BA+CA
A(B-C) = AB-AC
(B-C)A = BA-CA
a(B+C) = aB+aC
a(B-C) = aB-aC
(a+b)C = aC+bC
(a-b)C = aC-bC
a(bC) = (ab)C

mtrxac.zip :

* The properties of zero matrices.
* The properties of the transpose.
* The theorem of transpose.
* The theorem of inverse matrices.

mtrxad.zip :

* (A+B)**2.
* (A-B)**2.
* (A-B) (A+B).
* Power and inverse.
* Symetric and Skew-Symetric matrices.

mtrxae.zip :

* Solving linear systems by matrix inversions.
* Linear systems with common coefficient matrix.

mtrxaf.zip :

* The system of equation Ax = b is consistent.
* Inverses of symmetric matrices.
* Power, inverse of diagonal matrices
* Multiply, inverse of triangular matrices.
* Trace property.

mtrxag.zip : the value of the determinant of

* a diagonal matrix
* a triangular matrix (upper, lower)
* a basic matrix.

mtrxah.zip : LU decomposition :

* LU decomposition.
* Det(A) = Det(L).
* invA = invU invL.
* LU decomposition are not unique.

vectaa.zip Some vector space axioms on (rows,
columns) vectors and on polynomials.

* u + v = v + u
* (u + v) + w = u + (v + w)
* 0 + u = u + 0 = u
* u + (-u) = (-u) + u = 0
* k( u+ v) = ku + kv
* (k + l) u = ku + lu
* k (lu) = (kl) u

vectab.zip :

* Properties of Euclidian inner product in R**n.
* Properties of length in R**n.
* Properties of distance in R**n.
* u.v = 1/4 ||u+v||**2 - 1/4 ||u-v||**2.
* Cauchy-Schwarz inequality in R**n.
* If u.v =0 : ||u+v||**2 = ||u||**2 + ||v||**2.

vectac.zip : You can see the result in Gnuplot.

* Reflection about the x-axis.
* Reflection about the y-axis.
* Reflection about the line y = x.
* Orthogonal projection on the x-axis.
* Orthogonal projection on the y-axis.

vectad.zip : You can see the result in Gnuplot.

* Vector2d (vertical horizontal shift).

vectae.zip : You can see the result in Gnuplot.

* Reflection about the xy-plan.
* Reflection about the xz-plan.
* Reflection about the yz-plan.
* Orthogonal projection on the xy-plan.
* Orthogonal projection on the xz-plan.
* Orthogonal projection on the yz-plan.

vectag.zip :

* Linear combination in R**n, Pn.
* Linear combination of two vectors in R**n, Pn.
* Linear combination of three vectors in R**n, Pn.
* Vectors dependant or independant in R**n, Pn.
* Find the coordinate vector of (w)s in R**n, Pn.
* Find the coordinate vector of w. in R**n, Pn.

vectah.zip :

Inner product, norm, Distance in M22, Mnn, Mnm.

* Properties of Euclidian inner product
* Properties of distance.
* u.v = 1/4 ||u+v||**2 - 1/4 ||u-v||**2.

vectai.zip :

Inner product, norm, Distance on R**n generated by A..

* Properties of inner product on R**n generated by A.
* Properties of distance on R**n generated by A.
* u.v = 1/4 ||u+v||**2 - 1/4 ||u-v||**2.

vectaj.zip :

Weighed Euclidean Inner Product, norm, distance on R**n.

* Properties of Weighed Euclidean Inner Product on R**n.
* Properties of distance generated by
the Weighed Euclidean Inner Product on R**n.
* u.v = 1/4 ||u+v||**2 - 1/4 ||u-v||**2.

vectak.zip :

The Gram-Schmidt process.

* with the Euclidean inner product in R**n.
* with the inner product in M2x2.
* with the inner product in M3x2.
* with the inner product in M3x3.
* with the Inner Product on R**n generated by A.

vectal.zip :

Least squares, Orthogonal matrices

. * least squares solution of the linear system Ax = b.
* properties of Orthogonal matrices.

vectam.zip :

Eigenvalue, Eigenvector, Cayley-Hamilton theorem.

. * Some properties of Eigenvalue, Eigenvector.
* The properties of the Cayley-Hamilton theorem.

vectan.zip :

Some linear transformation. Is T a linear transformation?

* T(A) = AX.
* T(A) = trace(A).
* T(x1,x2,x3) = (x1 + x2 + x3).
* Matrix Inner Product on Mnxn.
* Euclidean Inner Product on R**n.
* Inner Product on R**n generated by A.
* Weighed Euclidean Inner Product on R**n.

vectao.zip :

Similarity

* B is similar to A.
* B is similar to A, also A is similar to B.
* A and invPAP have the same determinant.
* A is invertible if and only, if invPAP is invertible.
* A and invPAP have the same rank.
* A and invPAP have the same nullity.
* A and invPAP have the same trace.
* A and invPAP have the same, characteristic equation.
* The eigenvector of invPAP.
* If B is similar to A, then TrpsB and TrpsA are similar.

Application:

mtrxid.zip : Identity matrix application I

* Swap two rows.
* The pivot value.
* Eliminate the coefficient below, above, the pivot.
* Gauss Jordan elimination with the help of the identity matrix.
* Inverse of the matrix with the help of the identity matrix.

mtrxic.zip : Identity matrix application II

* The work on a column in one step.
* All the values below the pivot in one step.
* Application : Gauss elimination.
* All the values above the pivot in one step.
* Application : Gauss Jordan elimination
* Application : Inverse of the matrix

mtrxgo.zip : Geometric application.

You can verify the result into GnuPlot.

* Find the coefficients of a polynome,
that passes through three, four, five points.

* Find the coefficients a, b, c, d, e of a conic,
ax**2 + by**2 + cx + dy + e = 0
that passes through four points.

* Find the coefficients a, b, c, d of a circle,
a(x**2 + y**2) + bx + cy + d = 0
that passes through three points.

mtrxch.zip : Chemistry application.

* Find the coefficients of a chemical equation.

mtrxsy.zip : Resolve some nonlinear systems of equations.